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Mirrors > Home > MPE Home > Th. List > abeq1 | Structured version Visualization version GIF version |
Description: Equality of a class variable and a class abstraction. Commuted form of abeq2 2943. (Contributed by NM, 20-Aug-1993.) |
Ref | Expression |
---|---|
abeq1 | ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeq2 2943 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | |
2 | eqcom 2826 | . 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ 𝐴 = {𝑥 ∣ 𝜑}) | |
3 | bicom 224 | . . 3 ⊢ ((𝜑 ↔ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑)) | |
4 | 3 | albii 1814 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
5 | 1, 2, 4 | 3bitr4i 305 | 1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1529 = wceq 1531 ∈ wcel 2108 {cab 2797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 |
This theorem is referenced by: rabeqc 3676 disj 4397 dm0rn0 5788 dffo3 6861 dfsup2 8900 rankf 9215 fmla0xp 32623 dfon3 33346 dfiota3 33377 scottabf 40567 dffo3f 41428 |
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